How do you find the missing sides of a triangle given two angles and a side given Sides are x, y, and 10, two of the angles are 40 and 90?

1 Answer
marfre
May 2, 2017

If #10# = hypotenuse: #x~~6.43, y~~ 7.66#
If #10# is across from #50^@, x~~ 8.39, y ~~ 13.05#
If #10# is across from the #40^@, x~~ 11.92, y~~ 15.56# bolded text

Explanation:

Given: A triangle has two angles: #40^@, 90^@#, and side lengths #x, y, & 10#.

Since we have a #90^@# angle you know that you have a right triangle, which means you can use trigonometric functions.

There are three possible cases:

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The longest side is always across from the largest angle. This means that #10# is the hypotenuse if #10 is the longest side.

  1. Assume #10# is the hypotenuse:

Since a triangle's angles sum to #180^@#, the third angle is #50^@#.

#cos 40^@ = y/10; " " y = 10 cos 40^@ ~~ 7.66#

#sin 40^@ = x/10; " " x = 10 sin 40^@ ~~ 6.43#

  1. Assume #10# is across from the #50^@ # angle:

#tan 40^@ = x/10; " " x = 10 tan 40^@ ~~ 8.39#

#cos 40^@ = 10/y; " " y = 10/(cos 40^@) ~~ 13.05#

  1. Assume #10# is across from the #40^@ # angle:

#tan 40^@ = 10/x; " "x = 10/(tan 40^@) ~~ 11.92#

#sin 40^@ = 10/y; " " y = 10/(sin 40^@) ~~ 15.56#

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